The per capita disposable income for residents of a U.S. city in a recent year is normally distributed, with a mean of about \$44,000 and a standard deviation of about \$2450. Use this information in Exercises 7–10.


Out of 800 residents, about how many would you expect to have a disposable income of between \$40,000 and \$42,000?

Verified step by step guidance

1. Step 1: Identify the key parameters of the normal distribution. The mean (μ) is \$44,000, and the standard deviation (σ) is \$2,450. The problem asks for the proportion of residents with disposable incomes between \$40,000 and \$42,000.
2. Step 2: Convert the income values (\$40,000 and \$42,000) into z-scores using the formula: $z=\frac{x-\mu }{\sigma }$, where x is the income value, μ is the mean, and σ is the standard deviation.
3. Step 3: Look up the z-scores in the standard normal distribution table (or use a calculator) to find the cumulative probabilities corresponding to the z-scores for \$40,000 and \$42,000.
4. Step 4: Calculate the proportion of residents with incomes between \$40,000 and \$42,000 by subtracting the cumulative probability for \$40,000 from the cumulative probability for \$42,000.
5. Step 5: Multiply the proportion obtained in Step 4 by the total number of residents (800) to estimate the number of residents with disposable incomes in the specified range.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the per capita disposable income follows a normal distribution, which allows us to use statistical methods to estimate probabilities and expectations.

Z-Scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are essential for determining how many standard deviations an element is from the mean, which is crucial for finding probabilities in a normal distribution.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in estimating the proportion of residents with disposable incomes within a specific range, such as between $40,000 and $42,000 in this scenario.